In order to subtract one number from another, we place the subtrahend under the minuend, as follows: units under units, tens under tens. For example, let's take a two-digit number as a minuend, and a single-digit number as a subtrahend.

7 – 5 = 2 we write the result under the units.

Now we subtract tens from tens, but the subtrahend does not have tens, so we omit the ten of the reduced in response.

27 – 5 = 22

Now let's take both two-digit numbers:

Subtract the units of the subtrahend from the units of the minuend:

6 – 4 = 2 write the result under the units

Now subtract the tens of the subtrahend from the tens of the minuend:

8 – 3 = 5 we write the result under tens.

As a result, we get the difference:

86 – 34 = 52

Subtraction with the transition through the ten

Let's try to find the difference between the following numbers:

Subtract units. It is impossible to subtract 9 from 7, we take one ten from the tens of the reduced one. In order not to forget, we put a dot over the tens.

17 – 9 = 8

Now subtract tens from tens. The subtrahend has no tens, but we borrowed one ten from the minuend:

2 tens - 1 tens = 1 tens

As a result, we get the difference:

27 – 9 = 18

Now, for example, take three-digit numbers:

Subtract units. 2 less 8 , so we take one ten of the tens of the reduced one: 2 + 10 = 12 (we write 10 above the ones). In order not to forget, we put a dot over the tens.

12 – 8 = 4 the result is written under the units.

We occupied one ten of the tens for units, which means that in the reduced one there are no longer three tens, but two ( 3 tens - 1 tens = 2 tens).

Two tens less than six, take one hundred or 10 tens out of hundreds ( 2 tens + 10 tens = 12 tens write 10 over the tens of the minuend), and in order not to forget, we put an end to the hundreds. Subtract tens:

12 tens - 6 tens = 6 tens The result is written under the tens.

We occupied one hundred out of hundreds reduced for tens, which means we don’t have 9 hundreds, and 8 hundreds ( 9 hundreds - 1 hundred = 8 hundreds). Subtract hundreds:

8 hundreds - 7 hundreds = 1 hundred . We write the result under hundreds.

As a result, we get:

932 – 768 = 164

Let's complicate the task. What to do if in the category from which you need to take ten, is equal to zero? For instance:

We start with units. 2 less 8 , that is, it is necessary to take from tens. But for a decrease in tens 0 , which means that for tens you need to borrow from hundreds. In the hundreds place in the minuend too 0 , borrow from thousands. In order not to forget, we put a point over thousands.

In the hundreds of diminishing remains 9 , since we take one hundred for tens: 10 – 1 = 9 write 9 over hundreds.

Remains in the tens too 9 , since we took one ten for units: 10 – 1 = 9 write 9 over tens, and over units we write 10 .

Counting units:

12 – 8 = 4 write the result under the units.

Remaining in tens of minuends 9 , we consider:

9 – 6 = 3 write the result under tens.

Hundreds of diminishing left 9 , subtracted has no hundreds, omit 9 hundreds in response.

In the rank of thousands of diminished was 1 , we occupied it (dot over thousands), so there are no more thousands left. As a result, we get:

1002 – 68 = 934

So let's sum it up.

To find the difference between two numbers (column subtraction) :

1. we put the subtrahend under the minuend, we write units under units, tens under tens, and so on.
2. Subtract bit by bit.
3. If you need to take a ten from the next category, then put a dot over the category from which you borrowed. Above the category for which we occupy, we put 10.
4. If the digit from which we borrow is 0, then for it we borrow from the next digit of the reduced, over which we put a dot. Above the category for which they occupied, we put 9, since one ten was occupied.

How to subtract in a column

Subtraction of multi-digit numbers is usually performed in a column, writing the numbers one below the other (decreasing from above, subtracted from below) so that the digits of the same digits are one under the other (units under units, tens under tens, etc.). An action sign is placed between the numbers on the left. Draw a line under the subtrahend. The calculation begins with the discharge of units: units are subtracted from units, then from tens - tens, etc. The result of the subtraction is written under the line:

Consider an example when in some place the digit of the minuend is less than the digit of the subtrahend:

We cannot subtract 9 from 2, what should we do in this case? In the category of units, we have a shortage, but in the category of tens, the reduced one already has 7 tens, so we can transfer one of these tens to the category of units:

In the category of units, we had 2, we threw a dozen, it became 12 units. Now we can easily subtract 9 from 12. We write 3 under the line in the units place. In the tens place, we had 7 units, we threw one of them into simple units, 6 tens remained. We write under the line in the tens place 6. As a result, we got the number 63:

Subtraction by a column is usually not written down in such detail, instead, a dot is placed above the digit of the digit, from which the unit will be occupied, so as not to remember which digit will need to be additionally subtracted by the unit:

At the same time, they say this: you can’t subtract 9 from 2, we take a unit, we subtract 9 from 12 - we get 3, we write 3, we had 7 units in the tens place, we threw one, 6 left, we write 6.

Now consider column subtraction from numbers containing zeros:

Let's start subtracting. We subtract 3 from 7, write 4. We cannot subtract 5 from zero, so we are forced to take a unit in the highest digit, but we also have 0 in the highest digit, so for this digit we are also forced to take in a higher digit. We take a unit from the category of thousands, we get 10 hundreds:

We take one of the units of the hundreds digit to the least significant digit, we get 10 tens. Subtract 5 from 10, write 5:

In the hundreds place, we have 9 units left, so we subtract 6 from 9, write 3. In the thousands place, we had a unit, but we spent it on the lower digits, so zero remains here (you don’t need to write it down). As a result, we got the number 354:

Such a detailed record of the solution was given to make it easier to understand how subtraction by a column is performed from numbers containing zeros. As already mentioned, in practice the solution is usually written like this:

And all the mentioned actions are performed in the mind. To make subtraction easier, remember a simple rule:

If there is a dot above zero when subtracting, zero becomes 9.

Column Subtraction Calculator

This calculator will help you subtract numbers by a column. Just enter the minuend and subtrahend and click the Calculate button.

It is convenient to carry out a special method, which is called column subtraction or column subtraction. This method of subtraction justifies its name, since the minuend, the subtrahend and the difference are written in a column. Intermediate calculations are also carried out in columns corresponding to the digits of the numbers.

The convenience of subtracting natural numbers in a column lies in the simplicity of calculations. Calculations come down to using the addition table and applying the subtraction properties.

Let's see how column subtraction is performed. We will consider the subtraction process together with the solution of examples. So it will be clearer.

Page navigation.

What do you need to know to subtract by a column?

To subtract natural numbers in a column, you need to know, firstly, how subtraction is performed using the addition table.

Finally, it does not hurt to repeat the definition of the discharge of natural numbers.

Subtraction by a column on examples.

Let's start with the recording. The minuend is written first. Below the minuend is the subtrahend. Moreover, this is done in such a way that the numbers are one under the other, starting from the right. A minus sign is placed to the left of the recorded numbers, and a horizontal line is drawn below, under which the result will be recorded after the necessary actions have been taken.

Here are some examples of correct entries when subtracting by a column. Write down the difference in a column 56−9 , difference 3 004−1 670 , as well as 203 604 500−56 777 .

So, with the record sorted out.

We turn to the description of the process of subtraction by a column. Its essence lies in the sequential subtraction of the values ​​of the corresponding digits. First, the values ​​of the units digit are subtracted, then the values ​​of the tens digit, then the values ​​of the hundreds digit, and so on. The results are recorded under the horizontal line at the appropriate places. The number that is formed under the line after the completion of the process is the desired result of subtracting the two original natural numbers.

Imagine a diagram illustrating the process of subtraction by a column of natural numbers.

The above scheme gives a general picture of the subtraction of natural numbers by a column, but it does not reflect all the subtleties. We will deal with these subtleties when solving examples. Let's start with the simplest cases, and then we will gradually move towards more complex cases, until we figure out all the nuances that can occur when subtracting by a column.

Example.

First, subtract a column from the number 74 805 number 24 003 .

Solution.

Let's write these numbers as required by the column subtraction method:

We start by subtracting the values ​​​​of the digits of units, that is, we subtract from the number 5 number 3 . From the addition table we have 5−3=2 . We write the results obtained under the horizontal line in the same column in which the numbers are located 5 and 3 :

Now subtract the values ​​of the tens digit (in our example, they are equal to zero). We have 0−0=0 (we mentioned this property of subtraction in the previous paragraph). We write the resulting zero under the line in the same column:

Go ahead. Subtract the values ​​of the hundreds place: 8−0=8 (according to the property of subtraction, voiced in the previous paragraph). Now our entry will look like this:

Let's move on to subtracting the thousands place values: 4−4=0 (these are properties of subtraction of equal natural numbers). We have:

It remains to subtract the values ​​of the tens of thousands place: 7−2=5 . We write the resulting number under the line in the right place:

This completes the column subtraction. Number 50 802 , which turned out below, is the result of subtracting the original natural numbers 74 805 and 24 003 .

Consider the following example.

Example.

Subtract a column from the number 5 777 number 5 751 .

Solution.

We do everything in the same way as in the previous example - we subtract the values ​​of the corresponding digits. After completing all the steps, the entry will look like this:

Under the line we got a number in the record of which there are numbers on the left 0 . If these numbers 0 discard, then we get the result of subtracting the original natural numbers. In our case, we discard two digits 0 obtained on the left. We have: difference 5 777−5 751 is equal to 26 .

Up to this point, we have subtracted natural numbers whose records consist of the same number of characters. Now, using an example, let's figure out how natural numbers are subtracted in a column when there are more signs in the record of the reduced than in the record of the subtrahend.

Example.

Subtract from the number 502 864 number 2 330 .

Solution.

We write the minuend and the subtrahend in a column:

Subtract the values ​​of the unit digit one by one: 4−0=4 ; followed by tens: 6−3=3 ; further - hundreds: 8−3=5 ; further - thousand: 2−2=0 . We get:

Now, to complete the column subtraction, we still need to subtract the values ​​of the tens of thousands place, and then the values ​​of the hundreds of thousands place. But from the values ​​of these digits (in our example, from the numbers 0 and 5 ) we have nothing to subtract (since the subtracted number 2 330 does not have digits in these digits). How to be? Very simple - the values ​​​​of these bits are simply rewritten under the horizontal line:

On this subtraction by a column of natural numbers 502 864 and 2 330 completed. The difference is 500 534 .

It remains to consider the cases when, at some step of column subtraction, the value of the digit of the reduced number is less than the value of the corresponding digit of the subtrahend. In these cases, you have to "borrow" from the senior ranks. Let's understand this with examples.

Example.

Subtract a column from the number 534 number 71 .

Solution.

At the first step, subtract from 4 number 1 , we get 3 . We have:

In the next step, we need to subtract the values ​​of the tens digit, that is, from the number 3 subtract the number 7 . Because 3<7 , then we cannot perform the subtraction of these natural numbers (the subtraction of natural numbers is defined only when the subtrahend is not greater than the minuend). What to do? In this case, we take 1 unit from the highest order and "exchange" it. In our example, "exchange" 1 a hundred per 10 tens. To visually reflect our actions, we put a thick dot over the number in the hundreds place, and over the number in the tens place we write the number 10 using a different color. The entry will look like this:

We add received after the "exchange" 10 tens to 3 available tens: 3+10=13 , and subtract from this number 7 . We have 13−7=6 . This number 6 write under the horizontal line in its place:

Let's move on to subtracting the values ​​of the hundreds place. Here we see a dot above the number 5, which means that from this number we took one “for exchange”. That is, now we have 5 , a 5−1=4 . From number 4 nothing else needs to be subtracted (since the original subtracted number 71 does not contain digits in the hundreds place). Thus, under the horizontal line we write the number 4 :

So the difference 534−71 is equal to 463 .

Sometimes, when subtracting by a column, you have to “exchange” units from the highest digits several times. In support of these words, we analyze the solution of the following example.

Example.

Subtract from natural number 1 632 number 947 column.

Solution.

In the first step, we need to subtract from the number 2 number 7 . Because 2<7 , then you immediately have to "exchange" 1 dozen on 10 units. After that, from the sum 10+2 subtract the number 7 , we get (10+2)−7=12−7=5 :

In the next step, we need to subtract the tens digit values. We see that over the number 3 worth a point, that is, we have not 3 , a 3−1=2 . And from this number 2 we need to subtract the number 4 . Because 2<4 , then again you have to resort to "exchange". But now we are exchanging 1 a hundred per 10 tens. In this case, we have (10+2)−4=12−4=8 :

Now we subtract the values ​​of the hundreds place. From the number 6 unit was occupied in the previous step, so we have 6−1=5 . From this number we need to subtract the number 9 . Because 5<9 , then we need to "exchange" 1 a thousand per 10 hundreds. We get (10+5)−9=15−9=6 :

The last step remains. From the one in the thousands place we borrowed in the previous step, so we have 1−1=0 . We do not need to subtract anything else from the resulting number. This number is written under the horizontal line:

There is a convenient method for finding the difference of two natural numbers - subtraction in a column, or subtraction in a column. This method takes its name from the method of writing the minuend and the difference under each other. So you can carry out both basic and intermediate calculations in accordance with the required digits of numbers.

This method is convenient to use because it is very simple, fast and visual. All seemingly complex calculations can be reduced to addition and subtraction of prime numbers.

Below we'll look at exactly how to use this method. Our reasoning will be supported by examples for greater clarity.

What should be reviewed before learning column subtraction?

The method is based on some simple steps that we have already covered earlier. It is necessary to repeat how to subtract correctly using the addition table. It is also desirable to know the basic property of subtracting equal natural numbers (literally, it is written as a − a = 0). We will need the following equalities a − 0 = a and 0 − 0 = 0 , where a is any arbitrary natural number (if necessary, see the basic properties of finding the difference of integers).

In addition, it is important to know how to determine the digit of natural numbers.

The main thing at the first stage is to write down the initial data correctly. First, write down the first number from which we will subtract. Under it we place the subtrahend. The numbers must be located strictly one under the other, taking into account the category: tens under tens, hundreds under hundreds, units under units. The entry is read from right to left. Next, put a minus on the left side of the column and draw a line under both numbers. The final result will be written under it.

Example 1

Let's use an example to show which counting entry is correct:

With the help of the first, we can find how much 56 - 9 will be, with the help of the second - 3004 - 1670, the third - 203604500 - 56777.

As you can see, using this method, you can perform calculations of varying complexity.

Next, consider the process of finding the difference. To do this, we perform alternate subtraction of the values ​​of the digits: first, we subtract units from units, then tens from tens, then hundreds from hundreds, etc. The values ​​are written under the line separating the source data from the result. As a result, we should get a number, which will be the correct answer to the problem, i.e. the difference between the original numbers.

How exactly the calculations are performed can be seen in this diagram:

We figured out the general picture of recording and counting. However, there are some points in the method that need clarification. To do this, we will give specific examples and explain them. Let's start with the simplest tasks and gradually increase the complexity until we finally understand all the nuances.

We advise you to carefully read all the examples, because each of them illustrates separate incomprehensible points. If you reach the end and remember all the explanations, then the calculation of the difference of natural numbers in the future will not cause you the slightest difficulty.

Example 2

Condition: find the difference 74,805 - 24,003 using column subtraction.

Solution:

We write these numbers one under the other, correctly placing the digits under each other, and underline them:

Subtraction starts from right to left, that is, from units. We consider: 5 - 3 = 2 (if necessary, repeat the tables for adding natural numbers). We write the total under the line where the units are indicated:

Subtract tens. Both values ​​in our column are zero, and subtracting zero from zero always gives zero (remember, we mentioned that we will need this subtraction property later). The result is written in the right place:

The next step is to find the value of the thousand difference: 4 − 4 = 0 . The resulting zero is written to its proper place and as a result we get:

We got 50 802 , which will be the correct answer for the above example. This completes the calculations.

Answer: 50 802 .

Let's take another example:

Example 3

Condition: calculate how much will be 5 777 - 5 751 using the method of finding the difference by a column.

Solution:

The steps we need to take have already been given above. We execute them sequentially for new numbers and as a result we get:

The result is preceded by two zeros. Because they are the first, then you can safely discard them and get 26 in the answer. This number will be the correct answer of our example.

Answer: 26 .

If you look at the conditions of the two examples above, it is easy to see that so far we have taken only numbers that are equal in number of characters. But the column method can also be used when the minuend includes more characters than the subtrahend.

Example 4

Condition: find the difference 502 864 number 2 330 .

Solution

We write the numbers under each other, observing the desired correlation of digits. It will look like this:

Now we calculate the values ​​one by one:

– units: 4 − 0 = 4;

- tens: 6 - 3 \u003d 3;

– hundreds: 8 − 3 = 5;

- thousand: 2 − 2 = 0.

Let's write down what we got:

The subtrahend has values ​​in the place of tens and hundreds of thousands, but the minuend does not. What to do? Recall that emptiness in mathematical examples is equivalent to zero. So we need to subtract zeros from the original values. Subtracting zero from a natural number always gives zero, therefore, all that remains for us is to rewrite the original bit values ​​in the answer area:

Our calculations are complete. We got the total: 502 864 - 2 330 = 500 534 .

Answer: 500 534 .

In our examples, the values ​​of the digits of the subtrahend always turned out to be less than the values ​​of the minuend, so this did not cause any difficulties in the calculation. What if it is impossible to subtract the value of the bottom row from the value of the top row without going into a minus? Then we need to "borrow" the higher order values. Let's take a specific example.

Example 5

Condition: find the difference 534 - 71 .

We write the column already familiar to us and take the first step of calculations: 4 - 1 = 3. We get:

Next, we need to move on to counting tens. To do this, we need to subtract 7 from 3. This operation cannot be performed with natural numbers, because it only makes sense if the minuend is greater than the subtrahend. Therefore, in this example, we need to "borrow" a unit from the highest order and thereby "exchange" it. That is, we kind of change 100 for 10 tens and take one of them. In order not to forget about this, we mark the desired digit with a dot, and in tens we write 10 in a different color. We have a record like this:

The resulting result is written in the right place under the line:

It remains for us to finish the count by calculating hundreds. We have a point above the number 5: this means that we took ten from here for the previous digit. Then 5 − 1 = 4 . Nothing needs to be subtracted from the four, since the subtracted in the discharge of hundreds of values ​​\u200b\u200bhas no meaning. We write 4 in place and get the answer:

Answer: 463 .

Often, you have to perform the "exchange" action several times within one example. Let's take a look at this problem.

Example 6

Condition: how much is 1 632 - 947?

Solution

In the first stage of the calculation, it is necessary to subtract the two from the seven, so we immediately "occupy" the ten for exchange for 10 units. We mark this action with a dot and consider 10 + 2 - 7 = 5. Here's what our entry looks like with marks:

Next, we need to count the tens. The specified point means that for calculations we take a number one less in this bit: 3 − 1 = 2 . From the deuce, we have to subtract the four, so we "exchange" hundreds. We get (10 + 2) − 4 = 12 − 4 = 8 .

Moving on to counting hundreds. Of the six, we have already occupied one, so 6 − 1 = 5. We subtract nine from five, for which we take the thousand we have and "exchange" it for 10 hundred. So (10 + 5) − 9 = 15 − 9 = 6 . Now our note entry looks like this:

It remains for us to do the calculations in the thousandth place. We have already borrowed one unit from here, so 1 − 1 = 0 . We write the result under the final line and see what happens:

This completes the calculations. Zero at the beginning can be discarded. So 1632 − 947 = 685 .

Answer: 685 .

Let's take an even more complex example.

Today, in most cases, children master the simplest mathematical operations at preschool age. Parents try to teach their kids the basics of mathematics on their own, so that when they enter school, they already have a small but solid knowledge base. One skill that can be easily learned at home is counting.

Preparation for training

Before starting the study of counting in a column, parents need to make sure that the child is ready for classes. First of all, a young mathematician should easily count from 0 to 10 and easily distinguish all these numbers in writing. If the skill has not yet been consolidated or has not been mastered at all, it is imperative to fill in the gap. The most effective methods are presented in the article "".

In addition, the child should already understand the principles of simple mathematical operations, namely addition and subtraction. You should train daily, honing skills on nearby objects - toys, sweets, apples, counting sticks, etc. As soon as the child is confident enough to add and subtract single-digit numbers, you can move on to more complex tasks.

We count in a column

It is clear that adding and subtracting single-digit numbers in a column is meaningless - the child, as a rule, performs these actions in the mind. Difficulties arise when working with two-digit numbers - it is difficult for a beginner mathematician to concentrate and calculate everything without a visual representation. In this case, a method proven by several generations comes to the aid of the child - counting in a column.

Of course, mathematics teachers know how to teach a child to count with a column, but parents most often do not even know where to start classes. And you need to start from the base - an explanation of such a mathematical concept as bit depth. It is important for a child to understand how two-digit (and then three-digit) numbers are composed and how they are written when counting in a column. You can immediately perform a very simple but effective exercise - writing in a column of single and double digit numbers. The task of such an exercise is to teach the child to correctly place numbers with different digits under each other. The kid must understand that units are written under units, tens under tens, hundreds under hundreds, etc.

Having mastered this basic skill, the child can move on to the next stage - directly counting. It is necessary to explain to the baby that you need to add and subtract numbers by digits - units with units, tens with tens, hundreds with hundreds. Moreover, the account must be kept from units, that is, from right to left.

Some difficulties arise when adding numbers whose digits add up to more than "10", for example, 24 + 18. The child needs to be told that in this case the sum of units - "4" and "8" is "12". At the same time, under the units in the total amount, you also need to write only one, i.e. "2". And tens - "1" - must be "left in the mind." When adding dozens already - "2" and "1" in this example - it is necessary to add the ten "left in the mind", i.e. "1". As a result, adding tens looks like 2 + 1 + 1 and gives a total of "4". The final sum is "42". Similar actions must be performed when subtracting, when the digits of the minuend are less than the digits of the subtrahend. For example, 41 - 15. Only in this case it is necessary not to add the numbers “left in the mind”, but to subtract them.

So, in itself, the method of teaching a child to count in a column is quite understandable. But besides it, parents should familiarize themselves with general tips that will help make classes with the baby more effective:

• Be consistent and patient . Many adults believe that they are determined by age and the speed of mastering new educational material. However, forcing children to engage in an accelerated program is not worth it. You need to “grow up” to counting in a column, having first studied the basics, which have already been mentioned above.


• Repetition is the mother of learning. The success of classes depends on the amount of time devoted to practice. At every opportunity, turn to the child “for help” - ask him to count the numbers in a column and be sure to thank him when you get the result.


• Use additional materials . Children's books on mathematics, workbooks, diagrams and pictures will help children learn the material faster, because, as a rule, they perceive information presented visually better.


• Turn learning into play. This advice is universal for all training sessions. If you have the opportunity to include a game element in the learning process, the child will be more attentive and enthusiastic.

It is important to understand that the ability to count in a column does not determine. Therefore, you should not make high demands on the baby - he will definitely be able to independently perform mathematical operations in a column when he himself is ready for this.